TITLE:
Approach to Riemann Hypothesis by Combined Commensurable Step Function Approximation with Bonnet Method
AUTHORS:
Alfred Wünsche
KEYWORDS:
Riemann Zeta Function, Riemann Xi Function, Second Mean-Value Approach (Bonnet Method), Chebyshev Polynomials, Bessel Functions
JOURNAL NAME:
Advances in Pure Mathematics,
Vol.10 No.5,
May
6,
2020
ABSTRACT: To the Riemann hypothesis, we investigate first the approximation by step-wise Omega functions Ω(u) with commensurable step lengths u0 concerning their zeros in corresponding Xi functions Ξ(z). They are periodically on the y-axis with period proportional to inverse step length u0. It is found that they possess additional zeros off the imaginary y-axis and additionally on this axis and vanish in the limiting case u0 → 0 in complex infinity. There remain then only the “genuine” zeros for Xi functions to continuous Omega functions which we call “analytic zeros” and which lie on the imaginary axis. After a short repetition of the Second mean-value (or Bonnet) approach to the problem and the derivation of operational identities for Trigonometric functions we give in Section 8 a proof for the position of these genuine “analytic” zeros on the imaginary axis by construction of a contradiction for the case off the imaginary axis. In Section 10, we show by a few examples that monotonically decreasing of the Omega functions is only a sufficient condition for the mentioned property of the positions of zeros on the imaginary axis but not a necessary one.